Error Propagation - Estimating Variance

[unscientific]

Homework Statement

Not exactly a homework question, but rather a section in Statistical Data Analysis:

Suppose there is a pdf y_(x)[/SUB] that is not completely known, but μ_i and V_ij are known:

Homework Equations

The Attempt at a Solution

I understand how <y_(x)> ≈ y_(μ),

My confusion:

Why does <y_(x²)>

1. Imply we square everything throughout?

<[y_(μ) + Ʃ[∂y/∂x](x_i - μ_i)]²>

2. give a x_i and x_j term? Where did the x_j come from?

3. Why is it for i≠j when x_i and x_j are uncorrelated, the expression simplifies to

σ²_y ≈ Ʃ[∂y/∂x]²σ²_i

Where did the j go?

 

[Ray Vickson]

Homework Statement

Not exactly a homework question, but rather a section in Statistical Data Analysis:

Suppose there is a pdf y_(x)[/SUB] that is not completely known, but μ_i and V_ij are known:

Homework Equations

The Attempt at a Solution

I understand how <y_(x)> ≈ y_(μ),

My confusion:

Why does <y_(x²)>

1. Imply we square everything throughout?

<[y_(μ) + Ʃ[∂y/∂x](x_i - μ_i)]²>

2. give a x_i and x_j term? Where did the x_j come from?

3. Why is it for i≠j when x_i and x_j are uncorrelated, the expression simplifies to

σ²_y ≈ Ʃ[∂y/∂x]²σ²_i

Where did the j go?

_{It told you explicitly where the j "went": it said that $V_{ii} = \sigma_i^2$ and that $V_{ij} = 0$ for $i \neq j$.}

 

[unscientific]

It told you explicitly where the j "went": it said that $V_{ii} = \sigma_i^2$ and that $V_{ij} = 0$ for $i \neq j$.

Hmm, that makes sense.

What about the initial derivation? Why did they choose to square the entire RHS when it's a function of (x²) and not f²(x)? And the j's started appearing..

 

[Ray Vickson]

Hmm, that makes sense.

What about the initial derivation? Why did they choose to square the entire RHS when it's a function of (x²) and not f²(x)? And the j's started appearing..

Do you honestly mean to say that you cannot tell the difference between $g(x)^2$ and $g(x^2)$? The paper is working with $g(x)^2$!

 

[unscientific]

Do you honestly mean to say that you cannot tell the difference between $g(x)^2$ and $g(x^2)$? The paper is working with $g(x)^2$!

Ah I see, but where did the j's come from though? And nothing about y_(x²) was said that day.

 

[Ray Vickson]

Ah I see, but where did the j's come from though? And nothing about y_(x²) was said that day.

Try it for yourself: write out $(\sum_{i=1}^3 a_i)^2 = (a_1 + a_2 + a_3)^2$ in complete detail, by expanding out the square. After doing that, re-write the result using summation notation.

This will be my last post on this topic.